Embedding electronic structure in controllable quantum systems

ABSTRACT

Generating a computing specification to be executed by a quantum processor includes: accepting a problem specification that corresponds to a second-quantized representation of a fermionic Hamiltonian, and transforming the fermionic Hamiltonian into a first qubit Hamiltonian including a first set of qubits that encode a fermionic state specified by occupancy of spin orbitals. An occupancy of any spin orbital is encoded in a number of qubits that is logarithmic in the number of spin orbitals, and a parity for a transition between any two spin orbitals is encoded in a number of qubits that is logarithmic in the number of spin orbitals. An eigenspectrum of a second qubit Hamiltonian, including the first set of qubits and a second set of qubit, includes a low-energy subspace and a high-energy subspace, and an eigenspectrum of the first qubit Hamiltonian is approximated by a set of low-energy eigenvalues of the low-energy subspace.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.15/034,596, filed May 5, 2016, which is the U.S. National Stage ofInternational Application No. PCT/US2014/063825, filed on Nov. 4, 2014,published in English, which claims the benefit of U.S. ProvisionalApplication No. 61/900,119, filed on Nov. 5, 2013. The entire teachingsof the above application(s) are incorporated herein by reference.

STATEMENT AS TO FEDERALLY SPONSORED RESEARCH

This invention was made with government support under contractM1144-201167-DS awarded by the United States Department of Defense, andPHY-0955518 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

BACKGROUND

This invention relates to embedding electronic structure in controllablequantum systems.

Adiabatic quantum computing (AQC) works by changing the Hamiltonian of acontrollable quantum system from an initial Hamiltonian whose groundstate is easy to prepare into a Hamiltonian whose ground state encodesthe solution of a computationally interesting problem. The speed of thisalgorithm is determined by the adiabatic theorem of quantum mechanicswhich states that an eigenstate remains at the same position in theeigenspectrum if a perturbation acts on the system sufficiently slowly.Simply embedding a computational problem in a Hamiltonian suitable forAQC does not ensure an efficient solution. The required runtime for theadiabatic evolution depends on the energy gap between the ground stateand first excited state at the smallest avoided crossing.

AQC has been applied to classical optimization problems such as searchengine ranking, protein folding, and machine learning. There is anequivalence between a large set of such computational problems (problemsin the complexity class NP) and a set of models in classical physics(e.g., classical Ising models with random coupling strengths). For someAQC-based quantum computers, solutions of these computational problemsare based on the NP-Completeness of determining the ground state energyof classical Ising spin glasses. In general, quantum computing,including AQC, does not necessarily provide efficient solutions toNP-Complete problems in the worst case. However, there may exist sets ofinstances of some NP-Complete problems for which AQC can find the groundstate efficiently, but which defy efficient classical solution by anymeans.

Some quantum computers implemented as a controllable quantum systemsthat use some principles of AQC, but deviate from the requirement ofbeing strictly confined to the ground state at zero temperature and mayhave considerable thermal mixing of higher lying states. Such quantumcomputers are sometimes referred to as quantum annealing computers.Medium scale (e.g., 500 qubit) quantum annealing computers have beeninvestigated for many problems to determine if and by how much quantumannealing on the classical Ising model outperforms approaches usingoptimized codes on classical hardware for computing the same groundstate solution.

Another form of quantum computing is the gate model (also known as thecircuit model) of quantum computing. The gate model is based on ageneralization of the classical gate model where a classical bit (i.e.,a Boolean value of 0 or 1) is manipulated using logic gates. In thequantum gate model, a quantum bit or “qubit” (i.e., a quantumsuperposition of quantum basis states, such as a state representing a“0” and a state representing a “1”) is manipulated using quantum gates.While there is a form of computational equivalence for certaincomputations between the gate model and the AQC model, such that thecomputations can be performed in comparable amount of time using eithermodel, different problems can be mapped more easily into a computationmore suitable for one than the other. Also, the quantum systems thatrealize the gate model or the AQC model (including quantum annealingcomputers) may be very different. The molecular electronic structureproblem (also known as “quantum chemistry”) is an example of a problemthat has been mapped to the gate model, but not to the AQC model.

SUMMARY

In one aspect, in general, a specification compiler (e.g., softwareincluding instructions for causing a computer to perform atransformation procedure to transform a problem specification) is usedto configure a programmable computer, which then implements thetransformation procedure from a problem specification to a computingspecification suitable for execution on a quantum computer. The quantumcomputer can be implemented as a certain type of controllable quantumsystem on which a quantum annealing approach can be used to solve thespecified problem. As will be described in more detail below, approachesdescribed in this application address technical problems stemming fromone or both of: (1) requirements for expressing certain types of problemspecifications (e.g., molecular electronic structure problems) assuitable computing specifications, and (2) limitations on the form ofcomputing specification that may be executed by such quantum computers.Approaches described herein address these technical problems bytransforming an initial problem specification according to technicalconsiderations regarding how certain aspects of the problemspecification can be physically realized using the available types ofinteractions between hardware elements of the quantum computer thatrepresent qubits. These technical considerations are incorporated intothe compilation procedure used by the compiler for the quantum computer.

In another aspect, in general, a method, for use in transforming aproblem specification into a computing specification to be executed by aquantum processor that has limited types of couplings between hardwareelements representing quantum bits (qubits), includes: (1) accepting aproblem specification that corresponds to a second-quantizedrepresentation of a fermionic Hamiltonian associated with a model of amolecular system including one or more fermions and a particular numberof spin orbitals associated with the one or more fermions; (2)transforming the fermionic Hamiltonian into a first qubit Hamiltonianthat includes a first set of qubits that encode a fermionic statespecified by occupancy of the spin orbitals, where a transition betweentwo spin orbitals is associated with a parity of a sum of occupanciesacross spin orbitals between the two spin orbitals, the transformingincluding representing second-quantized fermionic operators within thefermionic Hamiltonian as interactions between qubits of the first set,where: (a) an occupancy of any spin orbital is encoded in a number ofqubits that is greater than one and less than a logarithmic function ofthe total number of spin orbitals, and (b) a parity for a transitionbetween any two spin orbitals is encoded in a number of qubits that isgreater than one and less than a logarithmic function of the totalnumber of spin orbitals; and (3) generating a second qubit Hamiltonianthat includes the first set of qubits and a second set of qubits, wherean eigenspectrum of the second qubit Hamiltonian includes a low-energysubspace characterized by a set of low-energy eigenvalues and ahigh-energy subspace characterized by a set of high-energy eigenvaluesthat don't overlap with the set of low-energy eigenvalues, and aneigenspectrum of the first qubit Hamiltonian is approximated by the setof low-energy eigenvalues.

Aspects may include one or more of the following features.

Perturbations used to generate the second qubit Hamiltonian excludesfrom the second Hamiltonian interactions in the first Hamiltonian thatviolate one or more constraints, and adds interactions between qubits ofthe second set.

The one or more constraints include constraining interactions betweenqubits in the second qubit Hamiltonian so that each interaction isbetween no more than two qubits.

The one or more constraints include constraining interactions betweenqubits in the second qubit Hamiltonian so that each interactioncorresponds to one of the limited types of couplings between hardwareelements of the quantum processor.

The limited types of couplings between hardware elements of the quantumprocessor constrain the interactions to exclude any interactionsrepresented by a term in the second qubit Hamiltonian that includes aPauli operator of a first type.

Excluding any interactions represented by a term in the second qubitHamiltonian that includes a Pauli operator of the first type includes:excluding any interactions represented by a term in the first qubitHamiltonian that comprises a product of an odd number of the Paulioperator of the first type.

The Pauli operator of the first type is a Pauli operator containing animaginary number.

The second-quantized fermionic operators within the fermionicHamiltonian includes fermionic annihilation and creation operators.

Transforming the fermionic Hamiltonian into a first qubit Hamiltonian isperformed using a Bravyi-Kitaev construction.

The first set of qubits corresponds to a first set of the hardwareelements, and the second set of qubits corresponds to a second set ofthe hardware elements.

The method further comprises configuring the quantum processor accordingto the computing specification.

The method further comprises: operating the quantum processor configuredaccording to the computing specification; and providing a specificationof a problem solution determined from a state of the quantum processorafter the operating.

Operating the quantum processor comprises operating said processoraccording to a quantum annealing procedure.

In another aspect, in general, a quantum computing system comprises aquantum processor configured to perform all the steps of any one of themethods above.

In another aspect, in general, software comprises instructions stored ona non-transitory computer-readable medium for causing a compiler toperform all the steps of any one of the methods above.

Aspects can have one or more of the following advantages.

The ability to make exact quantum chemical calculations on nontrivialsystems would revolutionize chemistry. While seemingly intractable forclassical algorithms, quantum computers can efficiently perform suchcomputations. There has been substantial interest in quantum algorithmsfor quantum chemistry using the gate model (involving Trotterization andquantum phase estimation). However, such gate-model approaches are notyet experimentally feasible for practical chemistry problems. Thetechniques described herein enable a different approach to quantumchemistry based on the quantum adiabatic algorithm. Approaches based onAQC, such as quantum annealing, do not require Trotterization, phaseestimation, or logic gates. Generally, the techniques described hereinenable scalable quantum simulation of fermionic systems using adiabaticquantum computing. Such simulation of fermionic systems enablesefficient computation of various molecular properties.

One of the technical challenges of mapping a quantum chemistry problemto an AQC model would be embedding the problem into the state of anappropriate controllable quantum system such as a quantum annealingcomputer. Embedding a problem to be solved into the state of acontrollable quantum system involves mapping the problem to aHamiltonian (i.e., an operator corresponding to the total energy of thesystem, whose eigenspectrum represents possible energy states of thesystem) that can be realized in the hardware elements of that quantumsystem. Also, mapping to such a realizable Hamiltonian requires carefulconsideration of the available properties of the quantum system. Oneproperty is the “control precision” of the Hamiltonian, which is thedynamic range of field values that must be resolved in order to measurethe intended eigenspectrum with a desired accuracy. This property isespecially important for molecular electronic structure Hamiltonians aschemists are typically interested in acquiring chemical accuracy (0.04eV). Control precision is often the limiting factor when a Hamiltoniancontains terms with coefficients that vary by several orders ofmagnitude. Other properties include the number of qubits available aswell as the connectivity and type of qubit couplings.

In some embodiments, a compiling procedure maps electronic structureHamiltonians to 2-body qubit Hamiltonians with a small set of physicallyrealizable couplings. As described in more detail below, combining theBravyi-Kitaev construction (which maps fermions to qubits, as describedin more detail below) with “perturbative gadgets” (which reduce theHamiltonian to 2-body interactions of particular types, as described inmore detail below) satisfies precision requirements on the couplingstrengths and provides an efficient mapping (i.e., a number of ancillaqubits that scale polynomially in the problem size). For someembodiments, the required set of controllable interactions includes onlytwo types of interaction beyond the Ising interactions required to applythe quantum adiabatic algorithm to combinatorial optimization problems.The mapping used by the compiler may also be of interest to chemistsdirectly as it defines a dictionary from electronic structure to spinHamiltonians with physical interactions, which can be used in a varietyof quantum computing applications.

Other features and advantages of the invention are apparent from thefollowing description, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of a computing system.

FIG. 2 is a flowchart of a procedure for operating the computing system.

FIGS. 3A-3D are diagrams of processes that occur in different examples.

FIG. 4 is an interaction graph for an embedded molecular HydrogenHamiltonian.

DESCRIPTION

Quantum chemistry applied to molecular systems is perhaps the broadestclass of problems on which quantum simulation of interacting fermionscould have an impact. Finding the energy of electrons interacting in theCoulomb potential of a set of fixed nuclei of an atom or moleculedefines the electronic structure problem. This problem iscomputationally expensive for classical computers because the cost ofdirectly solving for the eigenvalues of the exact electronic Hamiltoniangrows exponentially with the problem size.

One may divide quantum simulation algorithms into two classes: thosethat address statics and compute ground state properties, and those thataddress dynamics, and simulate time evolution of the wavefunction. It isclear that the simulation of time evolution is exponentially moreefficient on quantum computers, with significant implications for thesimulation of chemically reactive scattering. The computation of groundstate properties naturally requires preparation of the ground state.This can be done adiabatically, or by preparation of an ansatz for theground state. Adiabatic preparation of the ground state within a gatemodel simulation requires time evolution of the wavefunction, which isefficient. However, the length of time for which one must evolve isdetermined, as for all adiabatic algorithms, by the minimum energy gapbetween ground and first excited states along the adiabatic path. Thisis unknown in general. Similarly, a successful ansatz state must havesignificant overlap with the true ground state, and guarantees of thisare unavailable in general.

The worst case complexity of generic model chemistries (e.g. localfermionic problems studied with density functional theory) has beenshown to be in the quantum mechanical equivalent of NP-Complete,QMA-Complete. However, the subset of these generic models thatcorrespond to stable molecules, or to unstable configurations ofchemical interest such as transition states, is small and structured.Just as with adiabatic optimization, it does not matter if molecularelectronic structure is QMA-Complete so long as the average instance canbe solved (or even approximated) efficiently. In this case we also haveconsiderable heuristic evidence that molecules are able to find theirground state configurations rapidly: these are the configurations inwhich they naturally occur. Similarly, unstable transition states ofinterest occur in natural processes. Given that simulation of timeevolution on a quantum computer is efficient, we conjecture thatsimulation of the natural processes that give rise to these states willalso be practical.

The proofs that Local Hamiltonian (a decision problem capturing thecomplexity of finding the ground state energy) is QMA-Complete relies onthe construction of various specific Hamiltonians that can represent anypossible instance of any problem in QMA. In general, these Hamiltonianspossess couplings between more than two qubits. Hamiltonians thatcontain many-body interactions of order k and lower are referred to ask-local Hamiltonians; experimentally programmable couplings are 2-local.The original formulation by Kitaev was (log n)-local, he then reducedthis to 5-local and that result was subsequently reduced to 3-local. Toreduce 3-local Hamiltonians to 2-local Hamiltonians “perturbativegadgets” can be used to embed a k-local Hamiltonian in a subspace of a2-local Hamiltonian using ancilla qubits.

The compiler described herein uses a procedure that is scalable in thenumber of ancilla qubits, enabling the application of a quantumannealing algorithm (or other AQC-based algorithm) to a controllablequantum system that encodes a molecular electronic Hamiltonian. Thisscalability is useful because it limits the additional number ofhardware elements that may be needed in a quantum processor, asdescribed below. The problem specification provided as input to thecompiler may be in the form of a second quantized representation ofmolecular electronic structure. A second quantized representation is onein which the Hamiltonian is represented with fermionic creation andannihilation operators (i.e., also known as raising and loweringoperators for raising or lowering the energy of a quantum system,respectively).

In an example embodiment of the compilation procedure described hereinthree steps are used to convert an input problem specification to anoutput computing specification suitable for execution on a quantumcomputer. The first step includes converting the fermionic Hamiltonianof the input problem specification into a qubit Hamiltonian using theBravyi-Kitaev transformation. The Bravyi-Kitaev transformation is atechnique for representing second-quantized fermionic operators asinteractions between qubits that encode fermionic states by storing boththe occupancy and parity of each fermion non-locally, in alogarithmically upper-bounded number of qubits. The Jordan-Wignertransformation is another technique for representing second-quantizedfermionic operators as interactions between qubits. However, it isrecognized herein that use of the Bravyi-Kitaev transformation insteadof the Jordan-Wigner transformation is necessary for avoidingexponential control precision requirements in an a practical system. TheBravyi-Kitaev transformation was originally applied to solvingelectronic structure problems in the context of the gate model, asdescribed, for example, in a paper by Seely, J., Richard, M., and Love,P. entitled “The Bravyi-Kitaev transformation for quantum computation ofelectronic structure,” incorporated herein by reference. However, it isrecognized herein that it can be adapted for use in solving electronicstructure problems in the context of the AQC model.

The second step includes using a formulation of “perturbative gadgets,”which is a technique for embedding the eigenspectrum of a targetHamiltonian in the low energy subspace of a more constrained Hamiltonianby introducing a strongly gapped ancilla system and then choosing a setof constrained couplings to realize effective interactions as virtualtransitions between ancilla states that appear in a perturbativeexpansion for the effective low energy Hamiltonian. This techniqueallows removal all terms involving YY couplings in a single gadgetapplication (as used herein, X, Y and Z denote the Pauli operators, alsocalled Pauli matrices, and these operators are defined to act asidentity on unlabeled registers so that the dot product Y_(i)Y_(i) isunderstood to represent the tensor product Y_(i)⊗Y_(j)). By removingthese terms, the computing specification can be conformed to theavailable properties of the controllable quantum system. Some aspects ofthe particular perturbative gadgets technique used herein are describedin more detail in a paper by Cao, Y., Babbush, R., Biamonte, J., andKais, S. entitled “Towards Experimentally Realizable HamiltonianGadgets,” e-print arXiv:1311.2555 (Nov. 11, 2013), incorporated hereinby reference.

The third step includes applying “bit-flip” perturbative gadgets toreduce the locality of the Hamiltonian resulting in a 2-localHamiltonian with only ZZ, XX and ZX couplings. Aspects of this step aredescribed in greater detail below.

The following section describes an example of a system in which thecompiling techniques and resulting quantum computation techniquesdescribed herein can be used. The sections after that describe thetechnical considerations on which the compiling techniques are based inmore detail. In one section a second quantized formulation of theelectronic structure problem is described. The next section describesthe mapping of this problem to qubits. The next section describes thegadgets that we will use for interaction type and locality reduction.The next section includes a summary with description of variousalternative embodiments. Finally, an appendix includes an example of thecomputations performed by the compiling procedure for a simpleelectronic system of molecular hydrogen in a minimal basis.

Example System

Referring to FIG. 1, one context for solution of a problem uses aquantum computer 130. The quantum computer 130 has a number of hardwareelements (QB) 136 that are used to represent qubits of the quantumcomputation, and has coupling elements 137 that provide configurable andcontrollable coupling between the hardware elements 136, under thecontrol of a controller 134. The array of coupled hardware elements 136is referred to herein as a “quantum processor.” The hardware elements136 may be implemented, for example, using a circuitry capable ofstoring a quantum of a physical variable (e.g., magnetic flux) that isable to exist in a quantum superposition of two differentdistinguishable physical states (corresponding to a quantum state of anindividual qubit). The hardware elements 136 may include some elementsthat correspond to “logical” qubits and some elements that correspond to“ancilla” qubits, as described in more detail below. In someimplementations there are additional hardware components thatdistinguish the hardware elements for logical qubits from the hardwareelements for ancilla qubits. For example, the hardware elements for allof the qubits may include circuitry configured to provide a magneticfield in proximity to other circuitry of the qubit, and the hardwareelements for ancilla qubits may include circuitry that is configured toprovide a stronger magnetic field than the circuitry for the logicalqubits.

The controller 134 controls execution of a computing specification 132using the array of coupled hardware elements 136. The controller 134 maybe itself be implemented using any number of processors or other form ofclassical (i.e., non-quantum) programmable digital and/or analog controlcircuitry. The coupling elements 137 can be used to control theHamiltonian according to which the qubits evolve over time, and inparticular, can be used to control a time variation of the Hamiltonianin a quantum annealing procedure. After the quantum annealing procedure,physical quantities stored in the array of hardware elements 136 can bemeasured to yield a problem solution 140. In some embodiments, thehardware elements 136 include materials with particular properties suchas superconductors, and may be arranged into devices such as Josephsonjunctions. For example, a hardware element 136 may include a device thathas a superconducting loop interrupted by Josephson junctions known as asuperconducting quantum interference device (SQUID).

Generally, in some examples, this quantum computer 130 is used in thesolution of a problem, for example, a quantum chemistry problem, whosespecification 110 is provided by a user 190, or in some alternatives isprovided as the output of a system controller 180 (e.g., a computingsystem) that forms the problem specification 110 from anotherrepresentation of the problem. This problem specification 110 does not,however, necessarily meet the limitations of the computing specification132 that is able to be directly executed by the quantum computer 130. Inparticular, in this example, the quantum computer 130 requires acomputing specification 132 that is compatible with the types of quantuminteractions that are possible in the array of coupled hardware elements136.

The approaches described herein are based on transformations of theproblem specification 110 to form the computing specification 132 priorto control of the quantum computer 130 according to the computingspecification 132. These transformations may be considered somewhatanalogous to the compilation of a program written in a high-levelcomputing language to form a low-level (e.g., assembly language)specification, which may then be further processed (e.g., into a binaryexecutable file) before execution on a digital computer.

The transformation of the problem specification 110 to form thecomputing specification 132 is performed, in this example, on a digitalcomputer having a processor 126 as well as program memory, referred toin FIG. 1 as the specification compiler 120. The processing of theproblem specification 110 according to the procedures described in thisapplication makes use of software 122, which includes processorinstructions stored in the program memory (e.g., any non-transitorycomputer-readable medium) accessible by the processor 126. In someimplementations, the software 122 includes a mapping module 123 formapping the problem specification 110 to a qubit representation (e.g.,using the Bravyi-Kitaev transformation), and a reduction module 125 forreducing the initial qubit representation (e.g., using perturbativegadgets) to a form that enables the specification compiler 120 to outputa compatible computing specification 132.

In other examples, there may be additional system components and/orprocessing steps used to transform the output of the specificationcompiler 120 into a form appropriate for control of the hardwareelements 136 in the quantum computer, or for further processing theproblem solution 140 obtained from the time evolution of the quantumstate of the hardware elements 136. In the case in which the problemspecification 110 represents a quantum chemistry problem, the user 190and/or the system controller 180 may use the problem solution 140provided by the quantum computer 130 to determine chemical properties ofa molecule based on the electronic structure determined as the problemsolution 140.

Referring to FIG. 2, as well as FIG. 1, the overall process generallyfollows a number of steps. First, the problem specification 110 isaccepted (step 210) by the specification compiler 120, for example froma user 190 or a system controller 180. The mapping module 123 transformsa fermionic Hamiltonian into a first qubit Hamiltonian (step 220). Thereduction module 125 reduces the first qubit Hamiltonian to form (step230) a second qubit Hamiltonian from which the computing specification132 is generated. Having formed the transformed computing specification132, the quantum computer 130 is configured (step 240) according to thatcomputing specification 132. The quantum computer 130 is then operated(step 250) (under the control of the controller 134 according to thespecification 132) until the problem solution 140 is obtained.

Second Quantization

For determining the electronic structure of a desired atom or molecule,the system controller 180, based on input from the user 190, may providea problem specification 110 in the form of a Hamiltonian in a particularbasis, which can be used to express interactions of electrons with arepresentation using second-quantized fermionic operators (i.e.,annihilation and creation operators). The following is an example of afull configuration interaction (FCI) Hamiltonian in the occupationnumber basis. The example will be described in terms of variousquantities and symbols (e.g., corresponding to states or operators) tobe represented within the problem specification 110 in machine-readableform, and definitions will be provided to indicate what these quantitiesand symbols correspond to in terms of the quantum chemistry problembeing specified. We define spin orbitals as the product of a spinfunction (representing either spin up or spin down) and asingle-electron spatial function (usually molecular orbitals producedfrom a Hartree-Fock calculation). For example, in the case of molecularhydrogen there are two electrons and thus, two single-electron molecularorbitals, |ψ₁

and |ψ₂

. Electrons have two possible spin states, |α) (spin up) and |β

(spin down). The four spin orbitals for molecular hydrogen aretherefore, |χ₀

=|ψ₁

α

, |χ₁

=ψ₁

|β

, |χ₂

=|ψ₂

|α

, and |χ₃

=|ψ₂

|β

.

The occupation number basis is formed from all possible configurationsof n spin orbitals, which are each either empty or occupied by a singleelectron. We represent these vectors as a tensor product of individualspin orbitals expressed as |f_(n-1) . . . f₀) where f_(j)∈

indicates the occupation of spin orbital |χ_(j)

. Any interaction between electrons can be represented as somecombination of creation and annihilation operators α_(j) ^(†) and α_(j)for {j∈

|0≤j≤n}. Because fermionic wavefunctions must be antisymmetric withrespect to particle label exchange, these operators must obey thefermionic anti-commutation relations,[α_(j),α_(k)]₊=[α_(j) ^(†),α_(k) ^(†)]₊=0,[α_(j) ^(†),α_(k)^(†)]₊=δ_(jk)1  (1)With these definitions the second-quantized molecular electronicHamiltonian can be expressed as

$\begin{matrix}{H = {{\sum\limits_{i,j}\;{h_{ij}a_{i}^{\dagger}a_{j}}} + {\frac{1}{2}{\sum\limits_{i,j,k,l}\;{h_{ijkl}a_{i}^{\dagger}a_{j}^{\dagger}a_{k}{a_{l}.}}}}}} & (2)\end{matrix}$

The coefficients h_(ij) and h_(ijkl) are single and double electronoverlap integrals which may be precomputed classically. The number ofdistinct integrals scale as O(n⁴ in the number of spin orbitals n.

Qubit Representation

One aspect of the transformation performed by the specification compiler120 is to represent the fermionic wavefunction in terms of qubits, whichwill ultimately be mapped to the hardware elements 136. The following isa direct mapping that maps an occupancy state to a qubit basis state.Using Pauli operators we can represent qubit raising and loweringoperators as,Q _(j) ⁺=|1

0|=½(X _(j) −iY _(j)),Q _(j) ⁻=|0

1|=½(X _(j) +iY _(j)).

However, these operators do not obey the fermionic commutation relationsgiven in Eq. 1. To express qubit operators that obey the commutationrelations in Eq. 1, one option would be to use the Jordan-Wignertransformation.

However, the Jordan-Wigner transformation is not a scalable way toreduce electronic structure to an experimentally realizable Hamiltonianfor AQC. This is because the Jordan-Wigner transformation introducesk-local interaction terms into the Hamiltonian and k grows linearly inthe system size. While there are transformation procedures known as“perturbative gadgets” that allow for reductions in interaction order,using perturbative gadgets with the Jordan-Wigner transformation wouldlead to control precision increasing exponentially in k. Thus, thelinear locality overhead introduced by the Jordan-Wigner transformationwould translate into an exponential control precision requirement in thereduction.

An alternative mapping between the occupation number basis and qubitrepresentation, known as the Bravyi-Kitaev transformation, introduceslogarithmic locality overhead. By storing two pieces of information,creation and annihilation operators can be generated that act on qubitsand obey the fermionic commutation relations. First, the occupancy ofeach orbital is stored. Second, parity information is stored so that fora pair of orbitals, it is possible to determine the parity of theoccupancy of the orbitals that lie between them. This parity determinesthe phase that results from exchanging the occupancy of the twoorbitals.

The occupation number basis stores the occupation directly in the qubitstate (hence the name). This implies that occupancy is a fully localvariable in this basis; one may determine the occupancy of an orbital bymeasuring a single qubit. However, this also implies that the parityinformation is completely non-local. It is this fact that determines thestructure of the qubit creation and annihilation operators in theJordan-Wigner transformation. Each such operator changes the state of asingle qubit j (updating the occupancy information) but also acts on allqubits with indices less than j to determine the parity of theiroccupancy. This results in qubit operators, expressed as tensor productsof Pauli matrices, that contain strings of Z operators whose lengthgrows with the number of qubits. One could consider storing the parityinformation locally, so that the qubit basis states store sums oforbital occupancies. Then determination of parity requires a singlequbit operation. However, updating occupancy information requiresupdating the state of a number of qubits that again grows with thenumber of qubits. Hence this “parity basis” construction would offer noadvantage over the Jordan-Wigner transformation.

The Bravyi-Kitaev transformation offers a middle ground in which bothparity and occupancy information are stored non-locally, so neither canbe determined by measurement of a single qubit. Both parity andoccupancy information can be accessed by acting on a number of qubitsthat scales as the logarithm of the number of qubits. This logarithmicscaling makes the mapping of electronic structure to a 2-local qubitHamiltonian efficient.

Several subsets of qubits may be defined in which the parity andoccupancy information may be stored (non-locally). The occupancyinformation is stored in an “update set,” whereas the parity informationis stored in a “parity set.” These sets are distinct and their size isstrictly bounded above by the logarithm base two of the number ofqubits. The total number of qubits on which a qubit creation andannihilation operator may act can be a multiple of the logarithm basetwo of the number of qubits. However, this multiple is irrelevant fromthe point of view of the scalability of the construction.

Using the Bravyi-Kitaev transformation, the spin Hamiltonian formolecular hydrogen in the minimal (STO-3G) basis, may be expressed as

$\begin{matrix}{H_{H_{2}} = {{f_{0}1} + {f_{1}Z_{0}} + {f_{2}Z_{1}} + {f_{3}Z_{2}} + {f_{1}Z_{0}Z_{1}} + {f_{4}Z_{0}Z_{2}} + {f_{5}Z_{1}Z_{3}} + {f_{6}X_{0}Z_{1}X_{2}} + {f_{6}Y_{0}Z_{1}Y_{2}} + {f_{7}Z_{0}Z_{1}Z_{2}} + {f_{4}Z_{0}Z_{2}Z_{3}} + {f_{3}Z_{1}Z_{2}Z_{3}} + {f_{6}X_{0}Z_{1}X_{2}Z_{3}} + {f_{6}Y_{0}Z_{1}Y_{2}Z_{3}} + {f_{7}Z_{0}Z_{1}Z_{2}Z_{3}}}} & (4)\end{matrix}$where the integral values (in Hartree) are,f ₀=−0.81261,f ₁=0.17120,f ₂=0.16862,f ₃=−0.22278,f ₄=0.12055,f ₅=0.17435,f ₆=0.04532,f ₇=0.16587.  (5)

In general, the Bravyi-Kitaev transformation applied to electronicstructure produces an n-qubit Hamiltonian which is (log n)-local, andhas n⁴ real terms. This implies that each term has an even number of Yterms, or none, since the Pauli operators X and Z can be represented asmatrices with only real numbers as elements (zero values on one diagonaland real values on the other diagonal), and the Pauli operator Y can berepresented as a matrix containing imaginary numbers (zero values on themain diagonal and unit imaginary off-diagonal elements).

Hamiltonian Gadgets

Another aspect of the transformation performed by the specificationcompiler 120 is to ensure that the interactions between qubitsrepresented in the computing specification 132 are compatible with theavailable types of couplings that exist between the hardware elements136 physically representing those qubits. In order to embed electronicstructure in a Hamiltonian that is physically realizable in the quantumcomputer 130, the specification compiler 120 performs a scalableprocedure for transforming the (log n)-local qubit Hamiltonian into a2-local Hamiltonian with only ZZ, XX and XZ interaction terms (the termsthat correspond to realizable couplings between the hardware elements136). In this section, the procedure is described in terms of toolsknown as “Hamiltonian gadgets,” which allow the specification compiler120 to simulate the target Hamiltonian with these interactions.

Hamiltonian gadgets provide a method for embedding the eigenspectra (andsometimes eigenvectors) of an n-qubit “target” Hamiltonian, denoted byH_(target), in a restricted (typically low-energy) subspace of a moreconstrained (N>n)-qubit “gadget” Hamiltonian, denoted by {tilde over(H)}. To illustrate the general idea of gadgets, the following is anexample of how the specification compiler 120 can be configured to embeda k-local Hamiltonian into a 2-local Hamiltonian. Suppose that we have agadget Hamiltonian, {tilde over (H)}, which contains only 2-local termsthat act on N=n+a qubits (n “logical” qubits and a “ancilla” qubits).Then,

$\begin{matrix}{\left. {\left. {{\overset{\sim}{H} = {\sum\limits_{i = 1}{f_{i}O_{i}}}},{\overset{\sim}{H}❘\psi_{i}}} \right\rangle = {{\overset{\sim}{\lambda}}_{i}❘{\overset{\sim}{\psi}}_{i}}} \right\rangle,} & (6)\end{matrix}$where {f_(i)} are scalar coefficients, {tilde over (λ)}_(j) and |{tildeover (ψ)}_(i)

are the eigenvectors and eigenvalues of {tilde over (H)}, and {O_(i)}are the 2-local interaction terms of the physical Hamiltonian. Thespecification compiler 120 is configured to use interaction terms thatare Hilbert-Schmidt orthogonal so that Tr[O_(i)O_(j)]=2^(n)δ_(i,j). Wenow define an effective Hamiltonian that has support on the lowest 2^(n)states of the gadget Hamiltonian,

$\begin{matrix}{{H_{eff} \equiv {\sum\limits_{i = 0}^{2^{n} - 1}{{\overset{\sim}{\lambda}}_{i}\left. \overset{\sim}{\psi_{i}} \right\rangle\left\langle \overset{\sim}{\psi} \right.}}} = {\sum\limits_{i = 1}{f_{i}{O_{i} \otimes {\prod.}}}}} & (7)\end{matrix}$Here Π is a projector onto a particular state (usually the lowest energystate) of the a ancilla qubits and the {O_(i)} are a Hilbert-Schmidtorthogonal operator basis for operators on the space of the n logicalqubits. In other words, the most general representation of H_(eff) is anexpansion of all possible tensor products acting on the logical qubits.In general, there is no reason why f_(i)=0 on all non-2-local terms.Therefore a 2-local gadget on N=n+a qubits can embed a (k>2)-local,n-qubit Hamiltonian using a ancilla bits.

Perturbation theory techniques can be used enable the specificationcompiler 120 to generate Hamiltonian gadgets (due to QMA-Completeness of2-Local Hamiltonian). A construction referred to herein as the “bit-flipconstruction,” for reasons that will become clear below, is one suchperturbation theory technique. Different types of perturbative gadgetsmay have specific advantages for transforming Hamiltonians for specifictypes of hardware elements 136 or other hardware components of thequantum computer being used. Generally, there is a rough tradeoffbetween the number of ancillae required and the amount of controlprecision required. For example, bit-flip gadgets require less controlprecision than certain other gadget constructions (but generally moreancillae).

The following example analyzes the spectrum of the gadget Hamiltonian,{tilde over (H)}=H+V for the case that the norm of the perturbationHamiltonian, V, is small compared to the spectral gap between the groundstate and first excited state of the unperturbed Hamiltonian, H. Toaccomplish this we use the Green's function of {tilde over (H)}

$\begin{matrix}{{{\overset{\sim}{G}(z)} \equiv \left( {{z\; 1} - \overset{\sim}{H}} \right)^{- 1}} = {\sum\limits_{j}{\frac{\left. {\overset{\sim}{\psi}}_{j} \right\rangle\left\langle {\overset{\sim}{\psi}}_{j} \right.}{z - {\overset{\sim}{\lambda}}_{j}}.}}} & (8)\end{matrix}$We also define G(z) using the same expression except with H instead of{tilde over (H)}. Further, let H=L₊⊕L⁻ be the Hilbert space of {tildeover (H)} where L₊ is the “high-energy” subspace spanned by eigenvectorsof {tilde over (H)} with eigenvalues {tilde over (λ)}≥λ_(*) and L⁻ isthe complementary “low-energy” subspace, spanned by eigenvectors of{tilde over (H)} corresponding to eigenvalues of {tilde over (λ)}<λ_(*).Let Π_(±) correspond to projectors onto the support of L_(±). In arepresentation of H=L₊⊕L⁻, all the aforementioned operators V, H, {tildeover (H)}, G(z), {tilde over (G)}(z) are block-diagonal so we employ thenotation that A_(±±)=Π_(±)AΠ_(±) and,

$\begin{matrix}{A = {\begin{pmatrix}A_{+} & A_{+ -} \\A_{- +} & A_{-}\end{pmatrix}.}} & (9)\end{matrix}$Finally, we define the operator function known as the self-energy,Σ⁻(z)≡z1⁻ −{tilde over (G)} ⁻ ⁻¹(z).We use this notation to restate the “gadget theorem.”

Theorem 1 Assume that H has a spectral gap Δ around the cutoff λ_(*);i.e. all of its eigenvalues are in (−∞,λ⁻]∪[λ₊, +∞) where λ₊=λ_(*)+Δ/2and λ_(*)=λ_(*)−Δ/2. Assume that ∥V∥≤Δ/2. Let ε>0 be arbitrary. Assumethere exists an operator H_(eff) such that λ(H_(eff))⊂[c,d] for somec<d<λ_(*)−ε and, moreover, the inequality ∥Σ⁻ (z)−H_(eff)∥ε holds forall z∈[c−ε,d+ε]. Then each eigenvalue {tilde over (λ)}_(j) of {tildeover (H)}⁻ is ε-close to the j th eigenvalue of H_(eff).

Theorem 1 assures us that the eigenspectrum of the self-energy providesan arbitrarily good approximation to the eigenspectrum of the low-energysubspace of the gadget Hamiltonian. This is useful because theself-energy admits a series expansion,

$\begin{matrix}{{\Sigma_{-}(z)} = {H_{-} + V_{-} + {\sum\limits_{k = 2}^{\infty}{V_{- +}{G_{+}\left( {V_{+}G_{+}} \right)}^{k - 2}{V_{+ -}.}}}}} & (11)\end{matrix}$Using G₊=(z−Δ)⁻¹1₊ and H⁻=0, we focus on the range z=O(1)«Δ and findthat,

$\begin{matrix}{H_{cff} \approx {V_{-} + {\frac{1}{\Delta}{\sum\limits_{k = 2}^{\infty}{{V_{- +}\left( \frac{V_{+}}{\Delta} \right)}^{k - 2}{V_{+ -}.}}}}}} & (12)\end{matrix}$

The specification compiler 120 can be configured to use this effectiveHamiltonian to approximate the k-local target Hamiltonian, which we nowspecify. The terms in the target Hamiltonian will have a locality thatscales logarithmically with the number of spin orbitals. We may expresssuch a

$\begin{matrix}{T = {\overset{k - 1}{\underset{i = 0}{\otimes}}{O_{i}:{O_{i} \in {\left\{ {X_{i},Y_{i},Z_{i}} \right\}{\forall{i.}}}}}}} & (13)\end{matrix}$Gadgets can be applied term by term to reduce locality; however, thismay not be the optimal procedure. In addition, the specificationcompiler 120 is configured to replace even tensor powers of the Yoperator. A slightly more general form of term can be expressed as atarget for gadgetization. We use the fact that it is only the commutingnature of the {O_(i)} that is important for the gadget to function. Wemay therefore express a target term as a product of k commutingoperators, which includes the special case in which it is a product of koperators acting on distinct tensor factors,

$\begin{matrix}{T^{\prime} = {{\prod\limits_{i = 0}^{k - 1}{O_{i}:\left\lbrack {O_{i},O_{j}} \right\rbrack}} = {0{\forall\left\{ {i,j} \right\}}}}} & (14)\end{matrix}$Hence, we can represent the target Hamiltonian as a sum of r terms thatare the product of k commuting operators,

$\begin{matrix}{H_{target} = {H_{else} + {\sum\limits_{s = 1}^{r}{\prod\limits_{i = 0}^{k - 1}O_{s,i}}}}} & (15)\end{matrix}$where all {O_(s,i)} commute for a given s and H_(else) can be realizeddirectly by the physical Hamiltonian. While some formulations ofbit-flip gadgets have gadgetized operators acting on distinct tensorfactors, it is only necessary that the operators commute Their action ondistinct tensor factors is sufficient but not necessary for the gadgetconstruction. We take advantage of this property in order to transformYY terms appearing in the target Hamiltonian to other types of terms inthe gadget Hamiltonian. This transformation is useful, for example, inimplementations of the hardware elements 137 that do not permitcouplings between hardware elements 137 that directly implement qubitinteractions corresponding to YY terms. In particular, thistransformation can be accomplished by making the substitutionY_(i)Y_(j)→−X_(i)X_(j)Z_(i)Z_(j). Since X_(i)X_(j) commutes withZ_(i)Z_(j), we can create this effective interaction with a bit-flipgadget. For instance, suppose we have the term Z₀Y₁Y₂. We gadgetize theterm A·B·C where A=Z₀, B=−X₁X₂, and C=Z₁Z₂ and all operators A, B, Ccommute. Another approach to removing YY terms is explained in the papermentioned above by Cao, Y., Babbush, R., Biamonte, J., and Kais, S.entitled “Towards Experimentally Realizable Hamiltonian Gadgets,”e-print arXiv:1311.2555 (Nov. 11, 2013), incorporated herein byreference.

We now describe the form of the penalty Hamiltonian that acts only onthe ancilla qubits. Bit-flip gadgets introduce an ancilla system thathas two degenerate ground-states, usually taken to be |111 . . .

_(u) and |000 . . .

_(u) where u indicates that these kets refer to an ancilla space. Foreach of the r terms we use a separate ancilla system of the form,

$\begin{matrix}{{H_{s} = {\frac{\Delta}{2\left( {k - 1} \right)}{\sum\limits_{0 \leq i < j \leq {k - 1}}\left( {1 - {Z_{u_{s,i}}Z_{u_{s,j}}}} \right)}}}.} & (16)\end{matrix}$Again, we use u to indicate that operators act on an ancilla; e.g. thelabel u_(3,2) indicates the ancilla corresponding to O_(3,2) (the secondoperator in the third term). For each term we introduce an ancillasystem connected by a complete graph with equal and negative edgeweights. Thus, the ground state of the ancilla system is spanned by |111. . .

_(u) and |000 . . .

_(u).

Next, we introduce the perturbation Hamiltonian,

$\begin{matrix}{V = {H_{else} + \Lambda + {\mu{\sum\limits_{s = 1}^{r}{\sum\limits_{i = 0}^{k - 1}{O_{s,i}X_{u_{s,i}}}}}}}} & (17)\end{matrix}$where

$\mu = \sqrt[k]{\frac{\Delta^{k - 1}}{k!}}$and Λ is a 2-local operator on logical bits, which will be discussedlater. The effect of this Hamiltonian on the low energy subspace is tointroduce virtual excitations into the high energy space that modify thelow energy effective Hamiltonian. Only terms that start and end in theground state contribute to the perturbation series for the self-energy(see, for example, FIG. 3A). Thus, the gadget Hamiltonian will producethe target term at order k in which a transition between the twodegenerate ground states of the ancillae requires that each of the X_(u)terms in the perturbation act exactly once to flip all r·k bits from oneground state to the other. Crucially, the order in which the ancillaeare flipped does not matter since the operators O_(s,i) commute for agiven s. The complete gadget Hamiltonian is

$\begin{matrix}{\overset{\sim}{H} = {\Lambda + H_{else} + {\sum\limits_{s = 1}^{r}{\left\lbrack {{\mu{\sum\limits_{i = 0}^{k - 1}{O_{s,i}X_{u_{s,i}}}}} + {\frac{\Delta}{2\left( {k - 1} \right)}{\sum\limits_{0 \leq i < j \leq {k - 1}}\left( {1 - {Z_{u_{s,i}}Z_{u_{s,j}}}} \right)}}} \right\rbrack.}}}} & (18)\end{matrix}$and is related to the target Hamiltonian and effective Hamiltonian by{tilde over (H)} ⁻ =H _(target)⊗Π⁻ =H _(eff)  (19)for the appropriate choice of Λ and Δ»∥V∥ where Π⁻ projects onto theancillae ground space,Π⁻=|000

000|_(u)+|111

111|_(u).  (20)To illustrate the application of such a gadget and demonstrate how A ischosen, an example of a procedure to scalably reduce the locality ofmolecular hydrogen and remove all Y terms is provided in the nextsection.

For the example H_(target)=A·B·C+H_(else), the perturbation isV=μAX _(a) +αBX _(b) +μCX _(e) +H _(else)+Λ.  (21)Its components in the low energy subspace, as in the block diagonalrepresentation of Eq. 9 is:V ⁻=(H _(else)+Λ)⊗(|000

000|_(u)+|111

111|_(u)).  (22)

The projection into the high energy subspace is:

$\begin{matrix}{V_{+} = {{\left( {H_{else} + \Lambda} \right) \otimes \left( {\sum\limits_{{\{{a,b,c}\}} \in {\mathbb{B}}^{3}}{\left. {a,b,c} \right\rangle\left\langle {a,b,c} \right._{u}}} \right)} - V_{-} + {\mu\;{A \otimes \left( {{\left. {0,1,0} \right\rangle\left\langle {1,1,0} \right._{u}} + {\left. {1,1,0} \right\rangle\left\langle {{{0,1,0}}_{u} + {\left. {0,0,1} \right\rangle\left\langle {1,0,1} \right._{u}} + {\left. {1,0,1} \right\rangle\left\langle {0,0,1} \right._{u}}} \right)} + {\mu\;{B \otimes \left( {{\left. {1,0,0} \right\rangle\left\langle {1,1,0} \right._{u}} + {\left. {1,1,0} \right\rangle\left\langle {{{{{1,0,0}}_{u} + {\left. {0,0,1} \right\rangle\left\langle {0,1,1} \right._{u}} +}❘0},1,1} \right\rangle\left\langle {0,0,1} \right._{u}}} \right)}} + {\mu\;{C \otimes \left( {{\left. {1,0,0} \right\rangle\left\langle {1,0,1} \right._{u}} + {\left. {1,0,1} \right\rangle{\left\langle {{{1,0,0}}_{u} + {\left. {0,1,0} \right\rangle\left\langle {0,1,1} \right._{u}} + {\left. {0,1,1} \right\rangle\left\langle {0,1,0} \right._{u}}} \right).}}} \right.}}} \right.}}}} & (23)\end{matrix}$The projections coupling the low and high energy subspaces are:

$\begin{matrix}{V_{+ -} = {{\mu{A \otimes \left( {{\left. {1,0,0} \right\rangle\left\langle {0,0,0} \right._{u}} + {\left. {0,1,1} \right\rangle\left\langle {1,1,1} \right._{u}}} \right)}} + {{\mu B} \otimes \left( {{\left. {0,1,0} \right\rangle\left\langle {0,0,0} \right._{u}} + {\left. {1,0,1} \right\rangle\left\langle {1,1,1} \right._{u}}} \right)} + {\mu{C \otimes \left( {{\left. {0,0,1} \right\rangle\left\langle {0,0,0} \right._{u}} + {\left. {1\ ,1\ ,0} \right\rangle\left\langle {1\ ,1\ ,1} \right._{u}}} \right)}}}} & (24)\end{matrix}$and V⁻⁺=(V⁺⁻)^(†). Substituting these values into Eq. 12 we sec that atorder k=3 a term appears with the following form

$\begin{matrix}{{\frac{1}{\Delta^{2}}V_{- +}V_{+}V_{+ -}} = \left. {\frac{\mu^{3}}{\Delta^{2}}\left( {{ABC} + {ACB} + {BCA} + {CAB} + {BAC} + {CBA}} \right)}\rightarrow{AB{C.}} \right.} & (25)\end{matrix}$These terms arise because all ancilla qubits must be flipped and thereare six ways of doing so, representing 3! (in general this will be k!for a gadget Hamiltonian with k ancilla qubits) combinations of theoperators. These six terms are represented diagrammatically in FIG. 3A.Note that it is the occurrence of all orderings of the operators A, Band C that imposes the requirement that these operators commute Hence,in order to realize the desired term we see that

$\mu = {\sqrt[k]{\frac{\Delta^{k - 1}}{k!}}.}$A few competing processes occur that contribute unwanted terms, butthese terms either vanish with increasing spectral gap Δ, or they can beremoved exactly by introducing terms into the compensation term Λ. Oneway to compute Λ is to evaluate the perturbation series to order k andchoose Λ so that problematic terms disappear.

At higher orders we encounter “cross-gadget contamination,” which meansthat processes occur involving multiple ancilla systems, causingoperators from different terms to interact. For a 3-operator gadget,such terms will only contribute at order O(Δ⁻³). In reductions thatrequire going to higher orders, these terms do not necessarily depend onΔ, and so may introduce unwanted terms into the effective Hamiltonian.For instance, FIG. 3B shows an example of the four processes that occurat fourth order for a multiple term, 4-operator reduction. The diagramsinvolving multiple ancilla registers are examples of cross-gadgetcontamination.

However, if terms are factored into tensor products of operators thatsquare to the identity (as is the case for products of Pauli operators,which is always possible), cross-gadget contamination can onlycontribute a constant shift to the energy, which can be compensated forin Λ. This is because any process contributing to the perturbationseries that does not transition between the two different ground statesmust contain an even multiple of each operator, and if we choose to acton the non-ancilla qubits with operators that square to identity weobtain only a constant shift. Consider the two cross-gadget termsrepresented in these diagrams: A₁C₂ ²A₁=A₁1A₁=1 and D₂B₁D₂B₁=(D₂B₁)²=1.At even higher orders, individual cross-gadget terms might not equal aconstant shift (i.e., the sixth order term A₁A₂A₃A₂A₁A₃), but theoccurrence of all combinations of operators and the fact that all Pauliterms either commute or anti-commute will guarantee that such termsdisappear. In the sixth order example, if [A₁, A₂]=0 thenA₁A₂A₃A₂A₁A₃=A₁A₂A₃A₁A₂A₃=(A₁A₂A₃)²=1, otherwise [A₁, A₂]₊=0, whichimplies that A₁A₂A₃A₂A₁A₃+A₁A₂A₃A₁A₂A₃=0.

Example Problem: Molecular Hydrogen

The following is an example of a quantum chemistry problem that can besolved using the techniques described herein. We begin by showing howthe specification compiler 120 could be configured to factor andtransform the k-local molecular hydrogen Hamiltonian from Eq. 4 into a4-local part and a 2-local part so that H_(H) ₂ H_(4L)+H_(2L) where,

$\begin{matrix}{H_{4L} = {{\left( {{f_{4}Z_{0}} + {f_{3}Z_{1}}} \right)Z_{2}Z_{3}} + {\left( {Z_{1} + {Z_{1}Z_{3}}} \right)\left( {{f_{6}X_{0}X_{2}} + {f_{6}Y_{0}Y_{2}} + {f_{7}Z_{0}Z_{2}}} \right)}}} & (26) \\{\mspace{79mu}{H_{2L} = {{f_{0}1} + {f_{2}Z_{1}} + {f_{3}Z_{2}} + {f_{4}Z_{0}Z_{2}} + {f_{5}Z_{1}Z_{3}} + {f_{1}{{Z_{0}\left( {1 + Z_{1}} \right)}.}}}}} & (27)\end{matrix}$

In order to reduce H_(H) ₂ to a 2-local ZZ/XX/XZ-Hamiltonian thespecification compiler 120 further factors H_(4L) to remove YY terms,

$\begin{matrix}\begin{matrix}{H_{4L} = {{\underset{\underset{A_{1}}{︸}}{\left( {{f_{4}Z_{0}} + {f_{3}Z_{1}}} \right)}\underset{\underset{B_{1}}{︸}}{Z_{2}}\underset{\underset{C_{1}}{︸}}{Z_{3}}} + {\underset{\underset{A_{2}}{︸}}{f_{7}Z_{0}}\underset{\underset{B_{2}}{︸}}{Z_{2}}\underset{\underset{C_{2}}{︸}}{\left( {Z_{1} + {Z_{1}Z_{3}}} \right)}} +}} \\{\underset{\underset{A_{3}}{︸}}{f_{6}X_{0}X_{2}}\underset{\underset{B_{3}}{︸}}{\left( {1 - {Z_{0}Z_{2}}} \right)}\underset{\underset{C_{3}}{︸}}{\left( {Z_{1} + {Z_{1}Z_{3}}} \right)}} \\{= {{A_{1}B_{1}C_{1}} + {A_{2}B_{2}C_{2}} + {A_{3}B_{3}{C_{3}.}}}}\end{matrix} & (28)\end{matrix}$Within each term, the operators all commute so that[A_(i),B_(i)]=[A_(i),C_(i)]=[B_(i),C_(i)]=0. Factoring terms intocommuting operators is performed in order for bit-flip gadgets to workcorrectly.

Each of the operators defined in Eq. 28 will have a correspondingancilla qubit labelled to indicate the operator with which it isassociated, e.g. the ancilla for operator B₂ has label b₂. Theunperturbed Hamiltonian is a sum of fully connected ancilla systems inwhich each ancilla system corresponds to a term.

$\begin{matrix}{H_{1} = {{\frac{9\Delta_{1}}{4}1} - {\frac{\Delta_{1}}{4}{\left( {{Z_{a_{1}}Z_{b_{1}}} + {Z_{a_{1}}Z_{c_{1}}} + {Z_{b_{1}}Z_{c_{1}}} + {Z_{a_{2}}Z_{b_{2}}} + {Z_{a_{2}}Z_{c_{2}}} + {Z_{b_{2}}Z_{c_{2}}} + {Z_{a_{3}}Z_{b_{3}}} + {Z_{a_{3}}Z_{c_{3}}} + {Z_{b_{3}}Z_{c_{3}}}} \right).}}}} & (29)\end{matrix}$The spectral gap and Hamiltonian have the subscript “1” to associatethem with the first of two applications of perturbation theory. Theancilla system is perturbed with the Hamiltonian,

$\begin{matrix}{V_{1} = {{\mu_{1}\left( {{A_{1}X_{a_{1}}} + {B_{1}X_{b_{1}}} + {C_{1}X_{c_{1}}} + {A_{2}X_{a_{2}}} + {B_{2}X_{b_{2}}} + {C_{2}X_{c_{2}}} + {A_{3}X_{a_{3}}} + {B_{3}X_{b_{3}}} + {C_{3}X_{c_{3}}}} \right)} + H_{2L} + \Lambda_{1}}} & (30)\end{matrix}$where

$\mu_{1} = \sqrt[3]{\frac{\Delta_{1}^{2}}{6}}$and Λ₁ is a 2-local compensation Hamiltonian acting on the logicalqubits only. Later, Λ₁ will be selected by the specification compiler120 to cancel extraneous terms from the perturbative expansion. Theinteraction terms involving A, B, and C will arise at third order(V⁻⁺V₊V⁺⁻) from processes that involve a transition between the twodegenerate ground states of the ancilla systems. This occurs at thirdorder because to make the transition |000

|111

, we must flip all three ancilla bits in each term by applying theoperators X_(a), X_(b), and X_(c). Since these operators are coupled toA, B, and C, sequential action of bit flip operators yields theappropriate term. Because the operators commute, the order of the bitflipping does not matter. We now show how the specification compiler 120calculates the effective Hamiltonian using the perturbative expansion ofthe self-energy from Eq. 12.

Second Order

The only processes that start in the ground state and return to theground state at second order are those that flip a single bit and thenflip the same bit back. Thus, effective interactions are created betweeneach operator and itself,

$\begin{matrix}{{{- \frac{1}{\Delta_{1}}}V_{- +}V_{+ -}} = {{{- \frac{\mu_{1}^{2}}{\Delta_{1}}}\left( {A_{1}^{2} + B_{1}^{2} + C_{1}^{2} + A_{2}^{2} + B_{2}^{2} + C_{2}^{2} + A_{3}^{2} + B_{3}^{2} + C_{3}^{2}} \right)} = {{- \sqrt[3]{\frac{\Delta_{1}}{36}}}{\quad{\left\lbrack {{\left( {9 + f_{3}^{2} + f_{4}^{2} + f_{6}^{2} + f_{7}^{2}} \right)1} + {2f_{3}f_{4}Z_{0}Z_{1}} - {2Z_{0}Z_{2}} + {4Z_{3}}} \right\rbrack.}}}}} & (31)\end{matrix}$These processes are shown in FIG. 3C.The second order effective Hamiltonian at large Δ₁ is,

$\begin{matrix}{H_{eff}^{(2)} = {H_{2L} + \Lambda_{1} - {\sqrt[3]{\frac{\Delta_{1}}{36}}\left\lbrack {{\left( {9 + f_{3}^{2} + f_{4}^{2} + f_{6}^{2} + f_{7}^{2}} \right)1} + {2f_{3}f_{4}Z_{0}Z_{1}} - {2Z_{0}Z_{2}} + {4Z_{3}}} \right\rbrack} + {{O\left( \Delta_{1}^{- 2} \right)}.}}} & (32)\end{matrix}$

Third Order

The target Hamiltonian terms appears at third order from processes thattransition between degenerate ground states. However, there is also anadditional, unwanted process that occurs at this order. This competingprocess involves one interaction with H_(2L) and Λ₁ in the high-energysubspace,

$\begin{matrix}{{\frac{1}{\Delta_{1}^{2}}V_{- +}V_{+}V_{+ -}^{(1)}} = {{\frac{\mu_{1}^{2}}{\Delta_{1}^{2}}\left\lbrack {{{A_{1}\left( {H_{2L} + \Lambda_{1}} \right)}A_{1}} + {{B_{1}\left( {H_{2L} + \Lambda_{1}} \right)}B_{1}} + {{C_{1}\left( {H_{2L} + \Lambda_{1}} \right)}C_{1}} + {{A_{2}\left( {H_{2L} + \Lambda_{1}} \right)}A_{2}{B_{2}\left( {H_{2L} + \Lambda_{1}} \right)}B_{2}} + {{C_{2}\left( {H_{2L} + \Lambda_{1}} \right)}C_{2}} + {{A_{3}\left( {H_{2L} + \Lambda_{1}} \right)}A_{3}} + {{B_{3}\left( {H_{2L} + \Lambda_{1}} \right)}B_{3}} + {{C_{3}\left( {H_{2L} + \Lambda_{1}} \right)}C_{3}}} \right\rbrack}.}} & (33)\end{matrix}$These processes are illustrated diagrammatically in FIG. 3D.

The process we want occurs with the ancilla transition |000

|111

which flips all three bits (for each term separately since they havedifferent ancillae). There are 3!=6 possible ways to flip the bits foreach term, (these processes are illustrated in FIG. 3A),

$\begin{matrix}{{\frac{1}{\Delta_{1}^{2}}V_{- +}V_{+}V_{+ -}^{(2)}} = {{6\frac{\mu_{1}^{3}}{\Delta_{1}^{2}}\left( {{A_{1}B_{1}C_{1}} + {A_{2}B_{2}C_{2}} + {A_{3}B_{3}C_{3}}} \right)} = {{A_{1}B_{1}C_{1}} + {A_{2}B_{2}C_{2}} + {A_{3}B_{3}{C_{3}.}}}}} & (34)\end{matrix}$Because H_(2L) has no Δ₁ dependence and μ₁ is order O(Δ₁ ^(2/3), termssuch as (μ₁ ²/Δ₁ ²)A₁H 2LA₁ will vanish in the limit of large Δ₁;therefore, the third order effective Hamiltonian is,

$\begin{matrix}{H_{eff}^{(3)} = {H_{2L} + \Lambda_{1} - {\sqrt[3]{\frac{\Delta_{1}}{36}}\left\lbrack {{\left( {9 + f_{3}^{2} + f_{4}^{2} + f_{6}^{2} + f_{7}^{2}} \right)1} + {2f_{3}f_{4}Z_{0}Z_{1}} - {2Z_{0}Z_{2}} + {4Z_{3}}} \right\rbrack} + {\frac{\mu_{1}^{2}}{\Delta_{1}^{2}}\left( {{A_{1}\Lambda_{1}A_{1}} + {B_{1}\Lambda_{1}B_{1}} + {C_{1}\Lambda_{1}C_{1}} + {A_{2}\Lambda_{1}A_{2}} + {B_{2}\Lambda_{1}B_{2}} + {C_{2}\Lambda_{1}C_{2}} + {A_{3}\Lambda_{1}A_{3}} + {B_{3}\Lambda_{1}B_{3}} + {C_{3}\Lambda_{1}C_{3}}} \right)} + {A_{1}B_{1}C_{1}} + {A_{2}B_{2}C_{2}} + {A_{3}B_{3}C_{3}}}} & (35)\end{matrix}$with error O(Δ₁ ⁻³). We see that if

$\Lambda_{1} = {\frac{1}{\Delta_{1}}V_{- +}V_{+ -}}$then the unwanted contribution at third order will go to zero in thelimit of large Δ₁ and the second order term will cancel exactly with Λ₁.Thus,H _(eff) ⁽³⁾ ≈H _(2L) +A ₁ B ₁ C ₁ +A ₂ B ₂ C ₂ +A ₃ B ₃ C ₃  (36)H _(H) ₂ →H ₁ +V ₁  (37)where “→” denotes an embedding. There are still 3-local terms remainingin V₁,

$\begin{matrix}{V_{1} = {{{\mu_{1}\left( {{f_{4}Z_{0}} + {f_{3}Z_{1}}} \right)}X_{a_{1}}} + {\mu_{1}{X_{2}\left( {X_{b_{1}} + X_{b_{2}}} \right)}} + {\mu_{1}Z_{3}X_{c_{1}}} + {\mu_{1}f_{7}Z_{0}X_{a_{2}}} + {\mu_{1}{Z_{1}\left( {Z_{c_{2}} + X_{c_{3}}} \right)}} + {\mu_{1}X_{b_{3}}} + {\underset{\underset{A_{4}}{︸}}{\mu_{1}Z_{1}}\underset{\underset{B_{4}}{︸}}{Z_{3}}\underset{\underset{C_{4}}{︸}}{\left( {X_{c_{2}} + X_{c_{3}}} \right)}} + {\underset{\underset{A_{5}}{︸}}{\mu_{1}f_{6}X_{0}}\underset{\underset{B_{5}}{︸}}{X_{2}}\underset{\underset{C_{5}}{︸}}{X_{a_{3}}}} + {\underset{\underset{A_{6}}{︸}}{\left( {- \mu_{1}} \right)Z_{0}}\underset{\underset{B_{6}}{︸}}{Z_{2}}\underset{\underset{C_{6}}{︸}}{X_{b_{3}}}} + H_{2L} + {\Lambda_{1}.}}} & (38)\end{matrix}$With this notation we reorganize the Hamiltonian a final time, so thatH_(H) ₂ →H_(2L) H_(3L)

$\begin{matrix}{\mspace{79mu}{H_{3L} = {{A_{4}B_{4}C_{4}} + {A_{5}B_{5}C_{5}} + {A_{6}B_{6}C_{6}}}}} & (39) \\{H_{2L} = {{\left( {f_{0} + \frac{9\Delta_{1}}{4}} \right)1} + {f_{2}Z_{1}} + {f_{3}Z_{2}} + {f_{4}Z_{0}Z_{2}} + {f_{5}Z_{1}Z_{3}} + {f_{1}{Z_{0}\left( {1 + Z_{1}} \right)}} - {\frac{\Delta_{1}}{4}\left( {{Z_{a_{1}}Z_{b_{1}}} + {Z_{a_{1}}Z_{c_{1}}} + {Z_{b_{1}}Z_{c_{1}}} + {Z_{a_{2}}Z_{b_{2}}} + {Z_{a_{2}}Z_{c_{2}}} + {Z_{b_{2}}Z_{c_{2}}} + {Z_{a_{3}}Z_{b_{3}}} + {Z_{a_{3}}Z_{c_{3}}} + {Z_{b_{3}}Z_{c_{3}}}} \right)} + {\sqrt[3]{\frac{\Delta_{1}^{2}}{6}}\left\lbrack {{\left( {{f_{4}Z_{0}} + {f_{3}Z_{1}}} \right)X_{a_{1}}} + {Z_{3}X_{c_{1}}} + {f_{7}Z_{0}X_{a_{2}}} + {X_{2}\left( {X_{b_{1}} + X_{b_{2}}} \right)} + X_{b_{3}} + {Z_{1}\left( {X_{c_{2}} + X_{c_{3}}} \right)}} \right\rbrack} + {{\sqrt[3]{\frac{\Delta_{1}}{36}}\left\lbrack {{\left( {9 + f_{3}^{2} + f_{4}^{2} + f_{6}^{2} + f_{7}^{2}} \right)1} + {2f_{3}f_{4}Z_{0}Z_{1}} - {2Z_{0}Z_{2}} + {4Z_{3}}} \right\rbrack}.}}} & (40)\end{matrix}$The third order gadget used to reduce H_(3L) takes the same form asbefore except with the term labels 1, 2, 3 exchanged for the term labels4, 5, 6. The components of the final gadget Hamiltonian are

$\begin{matrix}{H_{2} = {{\frac{9\Delta_{2}}{4}1} - {\frac{\Delta_{2}}{4}\left( {{Z_{a_{4}}Z_{b_{4}}} + {Z_{a_{4}}Z_{c_{4}}} + {Z_{b_{4}}Z_{c_{4}}} + {Z_{a_{5}}Z_{b_{5}}} + {Z_{a_{5}}Z_{c_{5}}} + {Z_{b_{5}}Z_{c_{5}}} + {Z_{a_{6}}Z_{b_{6}}} + {Z_{a_{6}}Z_{c_{6}}} + {Z_{b_{6}}Z_{c_{6}}}} \right)}}} & (41) \\{\mspace{79mu}{and}} & \; \\{{V_{2} = {{\mu_{2}\left( {{A_{4}X_{a_{4}}} + {B_{4}X_{b_{4}}} + {C_{4}X_{c_{4}}} + {A_{5}X_{a_{5}}} + {B_{5}X_{b_{5}}} + {C_{5}X_{c_{5}}} + {A_{6}X_{a_{6}}} + {B_{6}X_{b_{6}}} + {C_{6}X_{c_{6}}}} \right)} + H_{2L} + \Lambda_{2}}}\mspace{79mu}{{{where}\mspace{14mu}\mu_{2}} = {\sqrt[3]{\frac{\Delta_{2}^{2}}{6}}\mspace{14mu}{and}}}} & (42) \\\begin{matrix}{\mspace{79mu}{\Lambda_{2} = {\frac{\mu_{2}^{2}}{\Delta_{2}}\left( {A_{4}^{2} + B_{4}^{2} + C_{4}^{2} + A_{5}^{2} + B_{5}^{2} + C_{5}^{2} + A_{6}^{2} + B_{6}^{2} + C_{6}^{2}} \right)}}} \\{= {{{\sqrt[3]{\frac{\Delta_{2}}{6}}\left\lbrack {\frac{7}{\sqrt[3]{6}} + {\Delta_{1}^{4/3}\left( {\frac{1}{3} + \frac{f_{6}^{2}}{6}} \right)}} \right\rbrack}1} + {\sqrt[3]{\frac{2\Delta_{2}}{9}}X_{c_{2}}{X_{c_{3}}.}}}}\end{matrix} & (43)\end{matrix}$This time the spectral gap and Hamiltonian have the subscript “2” toassociate them with the second application of perturbation theory. Wehave thus shown the embedding H_(H) ₂ →H₂+V₂. An interaction graph forthe embedded Hamiltonian is shown in FIG. 4.

Alternatives

We have presented a procedure for configuring the specification compiler120 for mapping any molecular electronic structure problem to a 2-localHamiltonian containing only ZZ, XX and XZ terms. The procedure isscalable in the sense that the computational resources (e.g., qubits,control precision, graph degree) scale polynomially in the number ofspin orbitals. The specification compiler 120 uses perturbative gadgetsthat embed the entire target Hamiltonian (as opposed to just the groundstate), thus guaranteeing that the eigenvalue gap is conserved under thereduction. Furthermore, the specification compiler 120 applies bit-flipgadgets to remove terms corresponding to couplings that are notavailable quantum computer, such as YY terms. The resulting Hamiltonianincluded in the computing specification 132 is suitable forimplementation in hardware elements 136 using techniques such assuperconducting systems, quantum dots, and other systems of artificialspins with the correct engineered interactions.

Further reduction of the types of interactions present in the computingspecification 132 is possible, to either ZZ and XX terms or ZZ and XZterms, for example. This makes the required interactions for simulatingelectronic structure Hamiltonians equivalent to the requirements ofuniversal adiabatic quantum computation. However, repeated reduction ofthe Hamiltonian results in more stringent precision requirements. Thechosen target set of interactions strikes a balance between controlprecision and a reasonable set of distinct types of controllableinteractions. The techniques described herein could also be applied tointeracting fermion problems on the lattice. However, in that case it ispossible to improve beyond the Bravyi-Kitaev mapping and exploit thelocality of the interactions to directly obtain Hamiltonians whoselocality is independent of the number of orbitals.

Any of a variety of techniques can be used to read out energyeigenvalues from the quantum computer 130, including using the tunnelingspectroscopy of a probe qubit. In this scheme, a probe qubit is coupledto a single qubit of the simulation. Tunneling transitions allow theprobe qubit to flip when the energy bias of the probe is close to aneigenvalue of the original system. Hence detection of these transitionsreveals the eigenspectrum of the original system. Thus, the eigenspectraof the molecular systems embedded into the spin Hamiltonian can bedirectly measured using the techniques described herein. Alternatively,the energy could be evaluated by determining the expectation value ofeach term in the Hamiltonian via projective measurements.

There exist classical algorithms, such as DMRG (density matrixrenormalization group) and related tensor network methods, and there arecomplexity and approximability results pertaining to minimal resourcemodel Hamiltonians. Using the techniques described herein, problems inchemistry can be reduced to such models, and solution of such problemscan be leveraged to make advances in electronic structure theory.

It is to be understood that the foregoing description is intended toillustrate and not to limit the scope of the invention, which is definedby the scope of the appended claims. Other embodiments are within thescope of the following claims.

What is claimed is:
 1. A method of compiling a quantum problem forexecution on a quantum processor, the method comprising: reading aproblem specification, the problem specification corresponding to afermionic Hamiltonian associated with a system of one or more fermion,the system having a plurality of spin orbitals; generating a firstcomputing specification from the problem specification, the firstcomputing specification comprising a first plurality of qubits, thefirst plurality of qubits encoding parity and occupancy information ofthe plurality of spin orbitals non-locally, wherein generating the firstcomputing specification comprises representing second-quantizedfermionic operators as interactions between qubits of the plurality ofqubits; generating a second computing specification from the firstcomputing specification, the second computing specification comprising aplurality of logical qubits and a plurality of ancilla qubits, whereingenerating the second computing specification comprises substitutinginteractions between ancilla qubits of the plurality of ancilla qubitsfor interactions between qubits of the plurality of qubits that areunavailable on the quantum processor.
 2. The method of claim 1, whereingenerating the first computing specification comprises applying aBravyi-Kitaev transformation to the Hamiltonian.
 3. The method of claim1, wherein the first plurality of qubits comprises a first subsetencoding parity information and a second, disjoint, subset encodingoccupancy information.
 4. The method of claim 3, wherein the firstsubset contains a number of qubits no greater than log₂ of a totalnumber of qubits in the first plurality of qubits.
 5. The method ofclaim 3, wherein the second subset contains a number of qubits nogreater than log₂ of a total number of qubits in the first plurality ofqubits.
 6. The method of claim 1, wherein the first computingspecification corresponds to a first Hamiltonian, the second computingspecification correspond to a second Hamiltonian, and the secondHamiltonian embeds the first Hamiltonian.
 7. The method of claim 6,wherein the second Hamiltonian is 2-local and the first Hamiltonian isgreater than 2-local.
 8. The method of claim 1, wherein said generatingthe second computing specification comprises removing couplingsunavailable on the quantum processor.
 9. The method of claim 1, wheresaid generating the second computing specification comprises removinginteractions that violate a locality constraint on the quantumprocessor.
 10. The method of claim 1, wherein the second-quantizedfermionic operators comprise annihilation and creation operators. 11.The method of claim 1, further comprising: executing the secondcomputing specification on the quantum processor.
 12. A non-transitorycomputer readable storage medium having program instructions embodiedtherewith, the program instructions executable by a processor to causethe processor to perform a method comprising: reading a problemspecification, the problem specification corresponding to a fermionicHamiltonian associated with a system of one or more fermion, the systemhaving a plurality of spin orbitals; generating a first computingspecification from the problem specification, the first computingspecification comprising a first plurality of qubits, the firstplurality of qubits encoding parity and occupancy information of theplurality of spin orbitals non-locally, wherein generating the firstcomputing specification comprises representing second-quantizedfermionic operators as interactions between qubits of the plurality ofqubits; generating a second computing specification from the firstcomputing specification, the second computing specification comprising aplurality of logical qubits and a plurality of ancilla qubits, whereingenerating the second computing specification comprises substitutinginteractions between ancilla qubits of the plurality of ancilla qubitsfor interactions between qubits of the plurality of qubits that areunavailable on the quantum processor.
 13. A system comprising: a quantumprocessor; a computer comprising a processor adapted to perform a methodcomprising: reading a problem specification, the problem specificationcorresponding to a fermionic Hamiltonian associated with a system of oneor more fermion, the system having a plurality of spin orbitals;generating a first computing specification from the problemspecification, the first computing specification comprising a firstplurality of qubits, the first plurality of qubits encoding parity andoccupancy information of the plurality of spin orbitals non-locally,wherein generating the first computing specification comprisesrepresenting second-quantized fermionic operators as interactionsbetween qubits of the plurality of qubits; generating a second computingspecification from the first computing specification, the secondcomputing specification comprising a plurality of logical qubits and aplurality of ancilla qubits, wherein generating the second computingspecification comprises substituting interactions between ancilla qubitsof the plurality of ancilla qubits for interactions between qubits ofthe plurality of qubits that are unavailable on the quantum processor;provide the second computing specification for execution on the quantumprocessor.